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February 4, 2005, 23:56 
FV matrices in terms of typical FE 'load vectors'

#1 
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Hi all,
I am currently researching both the FV Method & FE Method towards a PhD. I am attending lectures on each method  with different lecturers & hence different approaches. The approach used in the presentation of these methods appears to be very different: 1. FE Method  develops an understanding of conduction, convection matrices & associated 'load vectors'. Hence a lot of conceptual systembuilding is performed at a matrix level. whereas, 2. FV Method is presented differently, with sources seemingly accounting for any nonlinearity. No apparent concept of a matrixbuilding approach seems evident.  My question is to whether there are Academic texts which approach the FV Method from a similar angle to that of the matrixbuilding approach of FE Method. Ultimately, a system of matrices is built & solved. It appears that the nuances lie in the method of building the system matrices & load vectors. This topic is open for debate. Regards, Des Aubery... 

February 5, 2005, 09:51 
Re: FV matrices in terms of typical FE 'load vecto

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No takers yet?
I have been working on this approach for the past few days & am almost there. For me, at least, it helps to clarify both approaches in terms of the ultimate system algebraic relationships. I would value alternative texts for backchecking purposes... Regards, diaw... 

February 5, 2005, 13:58 
Re: FV matrices in terms of typical FE 'load vecto

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Well, I find that almost all FV texts tend to jump over the matrix building part. Usually they start with the generic PDE and then it automatically transforms into a system of A . x = b. I, as an engineer cannot understand certain transformations as well as a mathematician but I think there is the beauty of FV approach. For me the FV makes more sense to me since the discretization seems more related to physics and not so much math.
If you find anything simple and useful for an engineer please let me know. 

February 5, 2005, 20:36 
Re: FV matrices in terms of typical FE 'load vecto

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Thanks x_flow,
The problem with the FV A.x=b approach, as I see it, is precisely that all the effects are lumped into the matrix [A], and vector {b}. The FE approach splits the [A] matrix into [Kc]conduction & [Kh]convection  the {b} vector is split into a number of separate 'load' vectors eg {Qq} = internal heat generation vector, {Qh} = convection load vector, {Qc} = B/C load vector etc. When one uses this approach to 'build' the system from the element (or in this case control volume CV), one can investigate the effect of the various 'load vectors' on the system. It is also then a lot easier to modify etc. In my mind, it is a more elegant approach. In reality, FV method is closely linked to FD method & so we would have CV equations instead of element equations, but the logic appears to be similar. In the end, a sytem of equations is built & then analysed. The A.x=b lumping method tends  in my mind at least  make the problem more difficult to interrogate  especially for changes in various load conditions as one then has to generate NEW [A] & {b}, without much understanding of the direct contribution of the additional new effect. A few thoughts... Regards, diaw... 

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